RUI: Integrable Dynamics of Knotted Vortex Filaments

calini57在这个持续的项目中，我们探讨了涡旋丝方程的可积性的含义，涡旋丝方程是理想流体中细长涡旋丝自诱导运动的最简单模型，已知它等价于聚焦非线性薛定谔方程（NLS）。我们的研究目标是将大类别闭合细丝的几何和拓扑性质（特别是结类型）与相应周期性NLS解的Floquet谱联系起来，并了解细丝的结类型如何随着其演变而变化。我们采用的方法包括多相解的代数几何构造、Backlund变换、等周期变形和微扰理论。我们的工作是在微分方程和传统上属于纯数学的结理论之间建立联系的更大趋势的一部分，微分方程通常是物理现象的模型。在这里，我们研究了一个简化的涡运动模型，该模型在其解中具有丰富的结环类，并且其数学结构很好地理解，以便开发有效解决结形成，结结构稳定性和结分类等问题所需的数学工具。

NSF Award Abstract - DMS-0204557Mathematical Sciences: RUI: Integrable Dynamics of Knotted Vortex FilamentsAbstract0204557 CaliniIn this continuing project we explore the implications of integrability for the Vortex Filament Equation, which is the simplest model of the self-induced motion of a slender vortex filament in an ideal fluid, and which is known to be equivalent to the focusing nonlinear Schrodinger equation (NLS). The goals of our research are to relate geometric and topological properties -- in particular, knot types -- of large classes of closed filaments to the Floquet spectra of the corresponding periodic NLS solutions, and to understand how the knot type of a filament can change as it evolves. The methods we employ include algebro-geometric constructions of multi-phase solutions, Backlund transformations, isoperiodic deformations, and perturbation theory. Our work is part of a larger trend of making connections between differential equations, which usually model physical phenomena, and knot theory, which is traditionally a part of pure mathematics. Here, we study a simplified model of vortex motion, which possesses a rich class of knotted loops among its solutions, and whose mathematical structure is well understood, in order to develop the mathematical tools necessary to effectively address questions such as knot formation, stability of knotted structures, and knot classification.